A cord is wound round the circumference of wheel of radius r. The axis of the wheel is horizontal and moment of inertia about it is I. A weight mg is attached to the end of the cord and falls from the rest. After falling through a distance h, the angular velocity of the wheel will be
The Young's modulus of steel is twice that of brass. Two wires of same length and of same area of cross section, one of steel and another of brass are suspended from the same roof. If we want the lower ends of the wires to be at the same level, then the weights added to the steel and brass wires must be in the ratio of:
A particle, which is constrained to move along the x-axis, is subjected to a force in the same direction which varies with the distance x of the particle from the origin as \[F(x)=-kx+a{{x}^{3}}\]. Here k and a are positive constants. For \[x\ge 0,\] the functional form of the potential energy U(x) of the particle is
A solid sphereical conductor of radius R has a spherical cavity of radius a \[(a<R)\] at its centre. A charge \[+Q\] is kept at the centre. The cahrge at the inner surface, outer and at a position \[r(a<r<R)\] are respectively
Two straight horizontal parallel wires are carrying the same current in the same direction, d is the distance between the wires. You are provided with a small freely suspended magnetic needle. At which of the following positions will the orientation of the needle be independent of the magnitude of the current in the wires?
A)
At a distance \[d/2\] from any of the wires
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B)
At a distance \[d/2\] from any of the wires in the horizontal plane
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C)
Anywhere on the circumference of a vertical circle of radius d and centre halfway between the wires
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D)
At points halfway between the wires in the horizontal plane
Charges \[+q\] and \[-q\] are placed at points A and B respectively which are a distance 2L apart, C is the midpoint between A and B. The work done in moving a charge \[+Q\] from C to D along the semicircle CRD is
The density of a cube is measured by measuring its mass and length of its sides. If the maximum error in the measurement of mass and length are 4% and 3% respectively, the maximum error in the measurement of density will be
If a man throws up a ball and catches it after some time. This time is called time of flight. Time of flight in these types of motion depend on whether the man is stationary or he is accelerated. If the man is in a moving lift throws a ball and catches it after time \[{{t}_{1}}\] when lift is moving up with acceleration a and catches the ball in time \[{{t}_{2}}\] if the lift is moving down with same acceleration a. The speed with which ball is thrown w.r.t. man is
A stone projected with a velocity u at an angle \[\theta \] with the horizontal reaches maximum height \[{{H}_{1}}\]. When it is projected with velocity u at an angle \[\left( \frac{\pi }{2}-\theta \right)\] with the horizontal, it reaches maximum height \[{{H}_{2}}\]. The relation between the horizontal range R of the projectile, heights \[{{H}_{1}}\] and \[{{H}_{2}}\] is
The diagram shows the energy levels for an electron in a certain atom. Which transition shown represents the emission of a photon with the most energy?
In optical communication system operating at \[1200\text{ }nm,\] only \[2%\] of the source frequency is available for TV transmission having a bandwidth of\[5\text{ }MHz\]. The number of TV channels that can be transmitted is
A neutron travelling with a velocity v and kinetic energy E has a perfectly elastic head-on collision with a nucleus of an atom of mass number A at rest. The fraction of total energy retained by the neutron is approximately
A transverse wave is represented by the equation \[y={{y}_{0}}\sin \frac{2\pi }{\lambda }(vt-x)\]. For what value of \[\lambda \] is the maximum particle velocity equal to two times the wave velocity?
A vessel contains 110 g of water. The heat capacity of the vessel is equal to 10 g of water. The initial temperature of water in vessel is\[10{}^\circ C\]. If 220g of hot water at \[70{}^\circ C\] is poured in the vessel, the final temperature neglecting radiation loss, will be
Kinetic energy of a particle executing simple harmonic motion in straight line is \[p{{v}^{2}}\] and potential energy is \[q{{x}^{2}},\] where v is speed at distance x from the mean position. It's time period is given by the expression
Figure shows a parabolic graph between T and 1/V for a mixture of a gas undergoing an adiabatic process. What is the ratio of \[{{V}_{rms}}\] of molecules and to speed of sound in mixture?
In the Young's double-slit experiment, the intensity of light at a point on the screen where the path difference is \[\lambda \] is K, (\[\lambda \] being the wave length of light used). The intensity at a point where the path difference is \[\lambda /4,\] will be:
An electromagnetic wave of frequency \[v=3.0\text{ }MHz\] passes from vacuum into a dielectric medium with relative permittivity\[{{\varepsilon }_{r}}\]. Then
A)
wavelength is doubled and the frequency remains unchanged
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B)
wavelength is doubled and frequency becomes half
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C)
wavelength is halved and frequency remains unchanged
A metallic square loop ABCD is moving in its own plane with velocity v in a uniform magnetic field perpendicular to its plane as shown in the figure. An electric field is induced
If a ball of steel (density \[\rho =7.8\,g\,c{{m}^{-3}}\]) attains a terminal velocity of \[10\text{ }cm{{s}^{-1}}\] when falling in a tank of water (coefficient of viscosity \[{{\eta }_{water}}=8.5\times {{10}^{-4}}Pa-s\]) then its terminal velocity in \[(cm{{s}^{-1}})\] glycerine \[(\rho =12gc{{m}^{-3}},\eta =13.2Pa-s)\]would be nearly
A current of4A flows in a coil when connected to a \[12V\text{ }dc\]source. If the same coil is connected to a \[12V,50\text{ }rad/s\text{ }a.c.\] source, a current of \[2.4\,A\] flows in the circuit. Determine the inductance (in henry) of the coil.
Helium gas goes through a cycle ABCDA (consisting of two isochoric and isobaric lines) as shown in figure. Efficiency (in percent) of this cycle is nearly (Assume the gas to be close to ideal gas)
In a photocell bichromatic light of wavelength \[2475\text{ }\overset{o}{\mathop{A}}\,\]and \[6000\overset{o}{\mathop{A}}\,\] are incident on cathode whose work function is\[4.8\text{ }eV\]. If a uniform magnetic field of \[3\times {{10}^{-5}}\] tesla exists parallel to the plate, the radius (in cm) of the path described by the photoelectron is (mass of electron\[=9\times {{10}^{-31}}kg\])
When copper pyrites is roasted in excess of air, a mixture of CuO and FeO is formed. FeO is present as impurities. This can be removed as slag during reduction of CuO. The flux added to form slag is
In the hydrolytic equilibrium, \[{{A}^{-}}+{{H}_{2}}O\rightleftharpoons HA+O{{H}^{-}}\] \[{{K}_{a}}=1.0\times {{10}^{-5}}.\]
The degree of hydrolysis of a 0.001 M solution of the salt is
If the freezing point of a 0.01 molal aqueous solution of a cobalt (III) chloride-ammonia complex (which behaves as a strong electrolyte) is \[-0.0558{}^\circ C\], formula of complex is [\[{{K}_{f}}\]of water = 1.86 K kg \[\text{mo}{{\text{l}}^{-1}}\]]
Which of the given values is twice of the equivalent mass of the oxidising agent of the given reaction, \[S{{O}_{2}}+2{{H}_{2}}S\xrightarrow[{}]{{}}3S+2{{H}_{2}}O\]
The energy absorbed by each molecule \[({{A}_{2}})\]of a substance is \[4.4\times {{10}^{-19}}J\]and bond energy per molecule is \[4.0\times {{10}^{-19}}J\]. The kinetic energy of the molecule per atom will be
A metal (atomic mass \[=75gmo{{l}^{-1}}\] crystallizes in cubic lattice and the edge length of unit cell is \[5\overset{\text{o}}{\mathop{\text{A}}}\,\]. If the density of the metal is \[2gc{{m}^{-3}}\] then the radius of metal atom (in pm) is _______.
The rate constant for the first order decomposition of \[{{H}_{2}}{{O}_{2}}\]is given by the following equation: \[\log k=14.2-\frac{1.0\times {{10}^{4}}}{T}K\] The multiplication of \[{{E}_{a}}\]for this reaction and rate constant k if its half-life period is 200 minutes is ___.
A certain player, say X, is known to win with probability \[0.3\] if the track is fast and \[0.4\] if the track is slow For Monday probability of a fast track is \[0.7\] and probability of a slow track is \[0.3\]. The probability that player X will win on Monday is
Let \[f(x)={{\sin }^{2}}(x+\alpha )+{{\sin }^{2}}(x+\beta )-2\cos \] \[(\alpha -\beta )\sin (x+\alpha )\sin (x+\beta )\]. Then which of the following is TRUE?
A)
\[f(x)\] is strictly increasing in \[x\in (\alpha ,\beta )\]
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B)
\[f(x)\] is strictly decreasing in \[x\in (\alpha ,\beta )\]
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C)
\[f(x)\] is strictly increasing in \[x\in \left( \alpha ,\frac{\alpha +\beta }{2} \right)\] and strictly decreasing in \[x\in \left( \frac{\alpha +\beta }{2},\beta \right)\]
Given \[\Delta ABC\] with sides \[AB=6,\] \[AC=5,\] and \[CA=4.\]I is the incentre of \[\Delta ABC,\] and the internal angle bisectors of A B and C intersect the opposite sides.at D, E and F, respectively. The ratio of area of \[\Delta AIB\] to area of \[\Delta CID\] is
\[{{\sin }^{-1}}(\sin 1)+{{\sin }^{-1}}(\sin 2)+{{\sin }^{-1}}(\sin 3)+{{\sin }^{-1}}(\sin n)\] is a rational number, where n is a natural number. Number of such values of 'n' is
If complex number z lies on the curve \[|z-(-1+i)|=1,\] then the locus of the complex number \[\omega =\frac{z+i}{1-i},\] \[i=\sqrt{-1}\] is a circle having
A)
centre at \[(-3/2,1/2)\] and radius \[\frac{1}{\sqrt{2}}\]
doneclear
B)
centre at \[(3/2,-1/2)\] and radius \[\frac{1}{\sqrt{2}}\]
If \[\left| \left| \hat{a}+\hat{b}+\hat{a}\times \hat{b} \right|-\left| \hat{a}-\hat{b} \right| \right|=0,\] then the value of \[{{\left| \hat{a}-\hat{b} \right|}^{2}}\] is
The area of plane figure bounded by the lines \[y=\sqrt{x},\] \[x\in [0,1];\] \[y={{x}^{2}},\] \[x\in [1,2]\] and \[y=-{{x}^{2}}+2x+4,\]\[x\in [0,2]\] is ______.
Let \[f(x)={{x}^{4}}+|x|,\] \[{{I}_{1}}=\int\limits_{6}^{\pi }{f(\cos x)dx\,\,and\,\,{{I}_{2}}}=\int\limits_{0}^{\frac{\pi }{2}}{f(sin\,\,x)dx.}\] Then the value of \[\frac{{{I}_{2}}}{{{I}_{1}}}\] is _________.
If a twice differentiable function satisfies a relation \[f({{x}^{2}}y)={{x}^{2}}f(y)+yf({{x}^{2}})\forall x,y>0\] and \[f'(1)=1,\] then the value of \[f''(1/7)\] is ________.
Maximum length of chord of the ellipse \[\frac{{{x}^{2}}}{8}+\frac{{{y}^{2}}}{4}=1,\] such that eccentric angles of its extremities differ by \[\frac{\pi }{2}\] is____.